Optimal. Leaf size=84 \[ \frac{(b c-a d) (a+b x)^n (c+d x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n-1,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d^2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0297747, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {70, 69} \[ \frac{(b c-a d) (a+b x)^n (c+d x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n-1,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d^2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx &=\frac{\left ((-b c+a d) (a+b x)^n \left (\frac{d (a+b x)}{-b c+a d}\right )^{-n}\right ) \int (c+d x)^{-1-n} \left (-\frac{a d}{b c-a d}-\frac{b d x}{b c-a d}\right )^{1+n} \, dx}{d}\\ &=\frac{(b c-a d) (a+b x)^n \left (-\frac{d (a+b x)}{b c-a d}\right )^{-n} (c+d x)^{-n} \, _2F_1\left (-1-n,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d^2 n}\\ \end{align*}
Mathematica [A] time = 0.0319157, size = 83, normalized size = 0.99 \[ \frac{(b c-a d) (a+b x)^n (c+d x)^{-n} \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (-n-1,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d^2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{-1-n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{-n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{-n - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{-n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]